Absolute Value Equations
Equations with a variable or variables within absolute value bars are known as absolute value equations .
When solving equations that involve absolute values, there are two cases to consider.
Case 1: The expression inside the absolute value bars is positive.
Case 2: The expression inside the absolute value bars is negative.
For example, consider the expression .
For this to be true, either
You need to solve both equations. In this case, the solution to the first one is , and the the solution to the second one is . So there are two solutions , and .
It's also possible for an absolute value equation to have one solution :
has the single solution
or no solutions:
(The absolute value of any expression is positive, so there is no value of for which this is true.)